| L(s) = 1 | + (0.491 − 0.0867i)2-s + (−1.45 + 0.934i)3-s + (−1.64 + 0.598i)4-s + (9.76e−5 − 0.000553i)5-s + (−0.635 + 0.586i)6-s + (−2.32 − 1.26i)7-s + (−1.62 + 0.936i)8-s + (1.25 − 2.72i)9-s − 0.000280i·10-s + (−4.66 + 0.822i)11-s + (1.83 − 2.41i)12-s + (−1.77 + 2.10i)13-s + (−1.25 − 0.421i)14-s + (0.000375 + 0.000898i)15-s + (1.96 − 1.64i)16-s − 0.800·17-s + ⋯ |
| L(s) = 1 | + (0.347 − 0.0613i)2-s + (−0.841 + 0.539i)3-s + (−0.822 + 0.299i)4-s + (4.36e−5 − 0.000247i)5-s + (−0.259 + 0.239i)6-s + (−0.878 − 0.478i)7-s + (−0.573 + 0.331i)8-s + (0.417 − 0.908i)9-s − 8.87e − 5i·10-s + (−1.40 + 0.247i)11-s + (0.530 − 0.695i)12-s + (−0.490 + 0.585i)13-s + (−0.334 − 0.112i)14-s + (9.68e−5 + 0.000232i)15-s + (0.491 − 0.412i)16-s − 0.194·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00810226 + 0.205817i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00810226 + 0.205817i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.45 - 0.934i)T \) |
| 7 | \( 1 + (2.32 + 1.26i)T \) |
| good | 2 | \( 1 + (-0.491 + 0.0867i)T + (1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (-9.76e-5 + 0.000553i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (4.66 - 0.822i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (1.77 - 2.10i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + 0.800T + 17T^{2} \) |
| 19 | \( 1 - 5.20iT - 19T^{2} \) |
| 23 | \( 1 + (-1.01 + 1.21i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (5.40 + 6.44i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.76 - 7.58i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (1.02 + 1.78i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.20 - 4.36i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (6.39 + 2.32i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (6.96 + 2.53i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (9.11 - 5.26i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (10.1 + 8.49i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.24 + 3.41i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.369 + 2.09i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-7.56 - 4.37i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.22 - 3.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.447 - 2.53i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (1.47 - 1.23i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + 3.23T + 89T^{2} \) |
| 97 | \( 1 + (0.614 - 1.68i)T + (-74.3 - 62.3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.77234538674405968400472500915, −12.39871817141594026073549418145, −11.03586155525315483177671282663, −10.04127473417107384121666901572, −9.411716569833568169856651956748, −7.968939857215107533388335355529, −6.63325034811160704222737314265, −5.38089937921213682004077827180, −4.47876287123650789005771093627, −3.27371837768026646820302351243,
0.17659240921627613581416666065, 2.88559420094971775398608899728, 4.85476251356444731083411704659, 5.54076069893551182345969316532, 6.62061757200661074804374455740, 7.896459206942145630799875176582, 9.182920709645092501584475642797, 10.20716042661905357653628250988, 11.12571567727718635878545945996, 12.50603571401326710814074616147
